Matrix decomposition: Difference between revisions

Revision as of 23:04, 12 May 2021

Revision as of 08:23, 13 May 2021

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*Decomposition: <math>A=QS</math>, where ''Q'' is a complex orthogonal matrix and ''S'' is complex symmetric matrix.

*Uniqueness: If <math>A^\mathsf{T}A</math> has no negative real eigenvalues, then the decomposition is unique.<ref name=":0">{{Cite journal|last=Bhatia|first=Rajendra|date=2013-11-15|title=The bipolar decomposition|journal=Linear Algebra and Its Applications|volume=439|issue=10|pages=3031–3037|doi=10.1016/j.laa.2013.09.006|doi-access=free}}</ref>

*Comment: The existence of this decomposition is equivalent to <math>AA^\mathsf{T}</math> being similar to <math>A^\mathsf{T}A</math>.<ref>{{harvnb|Horn|merinoMerino|1995|pp=43–92}}</ref>

*Comment: A variant of this decomposition is <math>A=RC</math>, where ''R'' is a real matrix and ''C'' is a [[circular matrix]].<ref name=":0" />

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* [[Principal component analysis]]

== Notes References==

===Notes===

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===Citations===

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== References =Bibliography===

*{{cite journal|last1=Choudhury|first1=Dipa|last2=Horn|first2=Roger A.|title=A Complex Orthogonal-Symmetric Analog of the Polar Decomposition|journal=SIAM Journal on Algebraic and Discrete Methods|date=April 1987|volume=8|issue=2|pages=219–225|doi=10.1137/0608019}}

*{{citation|first=I.|last=Fredholm|title=Sur une classe d'´equations fonctionnelles|journal=Acta Mathematica|volume=27|pages=365–390|year=1903|language=fr|author-link=Ivar Fredholm|doi=10.1007/bf02421317|doi-access=free}}